Decimal magic squares

Achievement Objectives
NA3-1: Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals, and percentages.
Student Activity

Tui likes magic squares.
She decides to make all of the magic squares that she can using the numbers 2.0, 2.2, 2.4, 2.6 and 2.8

How many can she make if she uses each number at least once in the square?

It takes her quite a while because she doesn't know that the sum of a magic square is always three times the number in the centre.

Specific Learning Outcomes
Use addition with decimals
Know the idea of, and be able to construct, magic squares
Description of Mathematics

magic square is an arrangement like the one below where the vertical, horizontal and diagonal lines of numbers all add up to the same value. This ‘same value’ is called the sum of the magic square.

4
1
7
7 4 1
1 7 4

Magic squares are interesting objects in both mathematics proper and in recreational mathematics. It is likely that students have already encountered magic squares. The problems in this sequence give students the opportunity to use known numerical or algebraic concepts.

It’s a critical part of this and some later problems that three times the centre square is equal to the sum of the magic square.

This problem is part of a series exploring magic squares. The first of these is  Little Magic Squares  and A Square of Circles at Level 2,  Big Magic Squares also at Level 3. At Level 4, Negative Magic Squares, uses negative numbers and Fractional Magic Squares uses fractions. The Magic Square, Level 5 shows why three times the centre number is equal to the sum of the magic square. Finally, Difference Magic Squares at Level 6, looks at an interesting variation of the magic square concept.

Activity

The Problem

Tui likes magic squares. She decides to make all of the magic squares that she can using the numbers 2.0, 2.2, 2.4, 2.6 and 2.8

How many can she make if she uses each number at least once in the square?

It takes her quite a while because she doesn't know that the sum of a magic square is always three times the number in the centre.

Teaching sequence

  1. Talk about square ‘arrays’ of numbers like the ones in A Square of Circles . Ask the class if you can put numbers into these arrays so that the rows have the same sum; the columns have the same sum; all of the rows, columns and diagonals have the same sum.
  2. Show them a magic square such as the one below.
6 1 5
3 4 5
3 7 2
  1. Have them check that the rows all have the same sum (of 12); that the columns all have the same sum; and that the diagonals have the same sum. This is why they are called magic squares.
  2. Pose Tui’s problem.
  3. Have students work individually or in pairs to see how many magic squares they can find.
  4. As solutions emerge, ask some students to share and to prove that the arrays they have produced are magic squares?
  5. Pose the Extension problem as appropriate.
Extension to the Problem
Make up a magic square using at least four decimal numbers of your own choice. None of these magic squares can have the same number in each place.
 
Solution

Students are unlikely to solve the problem the way shown below.  Students are more likely to use guess and check and to stumble across the final set of solutions. However, a systematic approach is shown here to show that Tui should have found only one magic square.

You may however want to have your students see that there is a systematic way of finding the solution

Because the sum of the magic square is three times the center entry, consider the possible numbers that could go in the centre.
 
centre square = 2.0: Here the sum has to be 6. But the only way to get a sum of 6 from the numbers provided is to use 2.0 three times. But then we would have to have a magic square with 2.0 in every place. This is against the rules here.
 
centre square = 2.2: This forces us to have a sum of 6.6. This can only be done using 2.0, 2.2 and 2.4. We can get an answer here by replacing 7 by 2.0, 8 by 2.2 and 9 by 2.4 in the fourth answer in the Little Magic Square. Call this magic square A.
 
centre square = 2.4: Here the sum has to be 3 x 2.4. Now we can make 3 x 2.4 = 7.2 in a number of ways. These are by using 2.0, 2.4 and 2.8; 2.0, 2.6 and 2.6; 2.2, 2.2 and 2.8; 2.2, 2.4 and 2.6; and 2.4, 2.4 and 2.4. (These are the only possibilities. This can be shown by making a systematic list.)
Now we have to continue systematically.
First suppose that the main diagonal is 2.0, 2.4, 2.8. Then what can go into the centre top row square? It can’t be 2.0 (or 2.2) since there is no number that, along with 2.0 and 2.0 (2.2) gives 7.2. But it can be 2.4, 2.6 or 2.8. We take each of these in turn.
In the first two cases we get a problem where the question marks are. In the third case we get a magic square (B).
2.0 2.4 2.8
2.4 ??
2.8
2.0 2.6 2.6
2.4 ??
2.8
2.0 2.8 2.4
2.8 2.4 2.0
2.4 2.0 2.8
 
So now suppose that the main diagonal is 2.2, 2.4, 2.6. Then the top row centre square has to be 2.2, 2.4, 2.6 or 2.8. Here we seem to get three answers but the first and third of these are the same (flip the third one about the main diagonal). So we get two new magic squares here, C and D.
2.2 2.2 2.8
2.4 ??
2.6
2.2 2.4 2.6
2.8 2.4 2.0
2.2 2.4 2.6
2.2 2.6 2.4
2.6 2.4 2.2
2.4 2.0 2.6
2.2 2.8 2.2
2.4 2.4 2.4
2.6 2.0 2.6
   
This means that the only possibility left is 2.4, 2.4, 2.4 down the main diagonal. This leaves the five numbers 2.0, 2.2, 2.4, 2.6 and 2.8 as possibilities for the top row centre square. But putting 2.4 in there, we are forced to have all of the entries equal to 2.4, so we omit this possibility.
2.4 2.0 2.8
2.8 2.4 2.0
2.0 2.8 2.4
2.4 2.2 2.6
2.6 2.4 2.2
2.2 2.6 2.4
2.4 2.6 2.2
2.2 2.4 2.6
2.6 2.2 2.4
2.4 2.8 2.0
2.0 2.4 2.8
2.8 2.0 2.4
 
Now at first sight it looks as if we have found four more answers. However, the first and last one both have 2.0, 2.4 and 2.8 along a diagonal and so can be rotated to give the only answer with 2.0, 2.4 and 2.8 along the main diagonal. And the second and third are the same – just flip the third one about the main diagonal. So we end up with only one new answer E (the second one).
 
centre square = 2.6: This forces the sum of the magic square to be 7.8. The only way to get 7.8 here is to use 2.4, 2.6 and 2.8. This leads us to the new square F, which can be obtained from the 7, 8, 9 square in Little Magic Squares by changing 7 to 2.4, 8 to 2.6 and 9 to 2.8.
 
centre square = 2.8: This forces a sum of 8.4 but this can only be done with all entries equal to 2.8.
So we get 6 answers, A, B, C, D, E and F.

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